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78 Chapter 2. Examples of Banach and C�-algebras

2.2 The abstract Toeplitz algebra

2.2.1 Enveloping C�-algebra

Let A be a �-algebra, that is, an algebra equipped with an anti-linear, anti-multiplicative involution a ÞÑ a�. In general, the resulting notion of positivityfor elements in A can be pathological, for example by the existence of rela-tions of the form

°nk�1 a

�kak � 0. We will however be mostly interested in

�-algebras which admit a faithful representation on a Hilbert space.

Examples 2.2.1. 1. Let CrXs be the polynomial algebra in one variable,equipped with the unique �-algebra structure such thatX� � X. LetHbe a separable Hilbert space, and x P BpHq a self-adjoint element within�nite spectrum. Then we have an injective unital �-homomorphism

πx : CrXs Ñ BpHq, ppXq ÞÑ ppxq.

The closure of πxpCrXsq is the C�-algebra generated by idH and x, andhence isomorphic to CbpSppxqq.

2. Let Γ be a discrete group, and CrΓs the group algebra of Γ. Denote thenatural basis elements of CrΓs by λg. Then CrΓs is a �-algebra withthe �-structure uniquely determined by λ�g � λg�1 . We then have aninjective unital �-homomorphism, the regular representation,

πreg : CrΓs Ñ Bpl2pΓqq, λg ÞÑ ug,

where pugfqphq � fpg�1hq for f P l2pΓq. The closure of πregpCrΓsq, thatis, the C�-algebra generated by the ug, is called the reduced C�-algebraC�

redpΓq of Γ.

Another question one can pose for �-algebras is the following: given a �-algebra A, when does there exist a C�-algebra B and a �-homomorphismπu : AÑ B which is a solution to the universal problem of �-representing Ain C�-algebras? That is, pB, πuq should be such that for any C�-algebra Cand any �-homomorphism π : AÑ C, there exists a unique �-homomorphismΠ : B Ñ C such that Π � πu � π.

2.2 The abstract Toeplitz algebra 79

Bπ //

πu��

C

AΠ

?? (2.2)

In this case, we say A admits a universal C�-envelope, namely, the C�-algebraB. It is then unique up to �-isomorphism by abstract nonsense. Note thatwe do not require the associated homomorphism πu : AÑ B to be injective.

Lemma 2.2.2. Let A be a �-algebra with generating set S (so any elementin A is a polynomials expression in the elements of S YS�). Then A admitsa C�-envelope if and only if for every x P S, there exists Cx ¥ 0 such thatfor any �-representation π : AÑ BpHq, one has }πpxq} ¤ Cx.

Proof. Assume �rst that A admits a universal C�-envelope pB, πuq. Thenany �-homomorphism π : A Ñ BpHq extends to a �-representation Π : B ÑBpHq. As a �-representation of a C�-algebra is contractive, it follows that}πpxq} ¤ }πupxq} for any x P A.Conversely, assume for each x P S the existence of Cx ¥ 0 such that }πpxq} ¤Cx for all �-representations π. Then clearly such Cx's also exist for linearcombinations of products in the elements of S and S�. As S is generating,we deduce that such Cx's exist for all x P A.We may assume that the Cx are minimal with respect to this condition. Thenin fact for each x we can �nd a �-representation πx on a Hilbert spaceHx with}πxpxq} � Cx. Indeed, we can take πx � `nπn where πn is a

�-representationwith }πnpxq} ¥ Cx � 1

n.

De�ne then πu � `xPAπx on the Hilbert space H � `xPAHx. Let B be theclosure of πupAq. Then }πupxq} � Cx.

We claim that pB, πuq is the universal C�-envelope of A. Indeed, if C is aC�-algebra, we may assume that C � BpGq for some Hilbert space G. Ifthen π : AÑ C is a �-homomorphism, it follows that }πpxq} ¤ Cx � }πupxq}for every x P A. This implies that there exists a unique �-homomorphismΠ : B Ñ C such that Πpπupxqq � πpxq for all x P A.Examples 2.2.3. 1. Let A � CrXs with X � X�. Then A does not

admit a universal C�-algebra envelope. Indeed, for each self-adjoint


element x P BpHq, there exists a �-representation πx of A on H withπxpXq � x. Hence πx ÞÑ }πxpXq} is unbounded.

2. Let A be a Banach �-algebra. By Proposition 1.1.38, it follows that Aadmits a universal C�-envelope.

3. Let Γ be a discrete group. Then l1pΓq is a Banach �-algebra, cf. Ex-ample 1.1.36.5. It follows that l1pΓq admits a universal C�-envelopeC�

upΓq, called also the universal group C�-algebra of Γ. Note that wehave a factorisation

πred : C�u pΓq Ñ C�

redpΓq

to the reduced C�-algebra from Example 2.2.1.2. In general however,this map is not injective. We will come back to this in a later chapter.

Remark 2.2.4. One can also form the universal C�-algebra of a �-algebrawith given generators and a priori norm estimates. Namely, in Lemma 2.2.2one then only considers those representations in which the norm estimate issatis�ed. For example, the universal C�-envelope of CrXs with the norm-condition }X} ¤ 1 is the C�-algebra Cbpr�1, 1sq. Similarly, one can add theassumption of a priori positivity for elements.

2.2.2 The abstract Toeplitz C�-algebra

De�nition 2.2.5. The abstract Toeplitz �-algebra T is the unital �-algebragenerated7 by the single element s and the single relation s�s � 1.

Lemma 2.2.6. The abstract Toeplitz �-algebra T admits a universal C�-envelope.

Proof. Let π : T Ñ BpHq be a �-representation. Then p � πp1q is a self-adjoint projection, and moreover πpsq�πpsq � p. It follows from Lemma 2.2.2that T admits a universal C�-envelope.

De�nition 2.2.7. The abstract Toeplitz C�-algebra T is de�ned as the uni-versal C�-envelope of T .

7We recall that a unital �-algebra A is generated by a set B � A if any element of Acan be written as a polynomial expression in elements of t1u YB YB�.


Clearly, there exists a unital �-homorphism

T Ñ C�pSq � Bpl2pNqq, s ÞÑ S

into the concrete Toeplitz algebra. It then follows that it extends to a sur-jective �-homomorphism

T Ñ C�pSq.Our aim is to prove that this is in fact a �-isomorphism. This is equivalent toshowing that C�pSq is a universal solution to the C�-homomorphism problemfor T . This, in turn, is implied by the following.

Theorem 2.2.8 (Coburn's theorem). Let H be a Hilbert space, and U P Han isometry. Then there exists a unique unital �-representation

π : C�pSq Ñ BpHq, S ÞÑ U.

Moreover, if UU� � idH, then π is injective.

Proof. Of course unicity is clear, so it su�ces to show existence.

Let X be a set, and consider the Hilbert space HX � l2pX � Nq. Let SX bethe isometry

SX : HX Ñ HX , f ÞÑ pSXfq :

" pSXfqpx, 0q � 0pSXfqpx, nq � fpx, n� 1q, n ¥ 1.

Then under the natural identi�cation

HX � `xPX l2pNq,the operator SX corresponds to`xPXS. Hence we have a unital �-isomorphismC�pSq � C�pSXq sending S to SX .

On the other hand, let H be a Hilbert space and u a unitary. Then bythe spectral theorem, the inclusion Sppuq � S1 gives a �-homomorphismCbpS1q Ñ C�puq sending the identity function to u. Composed with thenatural map C�pSq Ñ CbpS1q, we obtain a �-representation C�pSq Ñ BpHqsending S to u.

Let now U be a general isometry. To prove the theorem, it su�ces to �nd aset X, a Hilbert space G, a unitary u P BpGq and a unitary

v : H Ñ l2pX � Nq ` G


such thatvUv� � SX ` u.

Such a decomposition is known as a Wold decomposition.

Let X be a set of cardinality dimpKerpU�qq (which by the above we mayassume ¡ 0, else U is a unitary and we're done). Fix an orthonormal basistex | x P Xu of KerpU�q. If n ¡ m and ξ, η P KerpU�q, then Unξ K Umη,since

xUnξ, Umηy � xξ, pU�qnUmηy� xξ, pU�qn�mηy� 0.

Hence we can de�ne an isometry

v1 : l2pX � Nq Ñ H, f ÞÑ¸

xPX,nPNfpx, nqUnex.

Let G be the orthogonal complement of the range of v1. Then

v : l2pX � Nq ` G Ñ H, ξ ` η ÞÑ v1ξ � η

clearly is a unitary.

We claim that G is invariant under U and U�, and that moreover the restric-tion u of U to G is a unitary. Indeed, we have η P G if and only if η K Unξfor any n ¥ 0 and any ξ P KerpU�q. Hence G is invariant under U and U�.By construction, KerpU�q X G � 0. It follows that also u� is an isometry,hence u a unitary.

It is now clear that v�Uv � SX ` u.

2.2.3 Quantum discs and �elds of C�-algebras

The Toeplitz C�-algebra T is generated by a single element S with SppSq �D, the closed unit disc. One might say that T is a non-commutative algebraof functions on the disc, or even more provocatively, an algebra of functionson a quantum disc. In the following, we are going to give some more meaningto this statement. First of all, let us show that one can interpolate betweenthe `quantum disc' and the real disc.


De�nition 2.2.9. Let 0 ¤ q 1. The quantum disc algebra at parameterq is the unital �-algebra Tq generated by the single element sq with the singlerelation

s�qsq � qsqs�q � 1� q.

We see that T0 � T .

Lemma 2.2.10. The quantum disc algebra Tq admits a universal C�-envelopeTq, and then }sq} � 1.

Proof. Let π be a (unital) �-representation of Tq, and put y � πpsqq. Theny�y � p1� qq � qyy�. It follows that (with rp�q the spectral radius)

}y}2 � }y�y}� rpy�yq� rpp1� qq � qyy�q� p1� qq � qrpyy�q� p1� qq � q}yy�}� p1� qq � q}y}2.

Hence p1 � qq}y}2 � p1 � qq, and }y} � 1. As Tq is generated by sq, thelemma follows.

The de�nition of Tq would of course also make sense for q � 1, which givesthe universal (commutative) �-algebra generated by a single normal elements1. However, this clearly does not admit a universal C�-algebra. But if weconsider the universal problem with respect to �-representations sending s1

to a contractive operator, then clearly the universal C�-algebra solution tothis problem is precisely the C�-algebra CbpDq.We aim to show now that in fact Tq � T for any 0 ¤ q 1. This is not truefor the underlying Tq.

Let us �rst �nd a concrete realization of Tq.

De�nition 2.2.11. Let 0 ¤ q 1, and let ten | n P Nu denote the standardorthonormal basis of l2pNq. De�ne a weighted shift on l2pNq by

Sq : l2pNq Ñ l2pNq, en ÞÑa

1� qn�1en�1.

We de�ne the concrete quantum disc C�-algebra at parameter q to be C�pSqq.


It is clear that S�q e0 � 0 and

S�q en �a

1� qnen�1, n ¡ 0,

and that Sq satis�es the quantum disc algebra relation

S�q Sq � qSqS�q � p1� qqidl2pNq.

Proposition 2.2.12. One has C�pSqq � C�pSq, and πQpSqq � πQpSq in theCalkin algebra.

Proof. Let Tq � 1� S�q Sq. Then Tqen � qn�1en, hence Tq P B0pl2pNqq. SinceSq � Sp1� Tqq1{2, it follows that πQpSq � πQpSqq.Now B0pl2pNqq � C�pSq, hence Sq P C�pSq and so C�pSqq � C�pSq. Con-versely, since Tq P C�pSqq and p1 � Tqq1{2 invertible, we deduce S P C�pSqq.We conclude C�pSq � C�pSqq.Theorem 2.2.13. All Tq with 0 ¤ q 1 are �-isomorphic C�-algebras.

Proof. Let us show that the canonical map πq : Tq Ñ C�pSqq sending sq toSq is a

�-isomorphism. The theorem will then follow by Proposition 2.2.12.

By Coburn's theorem, we may suppose that q ¡ 0. By the non-commutativeGelfand-Neumark theorem, we may also assume that Tq � BpHq for someHilbert space H. Write tq � 1� s�qsq. We then claim that Spptqq � r0, qs.Indeed, let us �rst show that Sppsqs�q q � t0, 1� q, 1� q2, . . .uY t1u. Supposethis were not the case. Then we can �nd λ P Sppsqs�q q with λ � 1 andλ � 1 � qn for all n P N. But since Sppxyq Y t0u � Sppyxq Y t0u for anycouple of operators x, y, we deduce from the de�ning relation for sq that alsoq�1pλ � p1 � qqq is a non-zero element in Sppsqs�q q. By induction, we obtainthat q�npλ � 1q � 1 P Sppsqs�q q for all n P N. But q�npλ � 1q � 1 0 for nlarge enough, giving a contradiction with Sppsqs�q q � r0,�8q.Hence also Spps�qsqq � t0, 1�q, 1�q2, . . .uYt1u. Now if 0 P Spps�qsqq, it followsfrom the de�nining relation of sq that p1� q�1q P Sppsqs�q q, contradicting thepositivity of the operator sqs

�q . Hence Spps�qsqq � t1� q, 1� q2, . . .u Y t1u.

We deduce that indeed Spptqq � r0, qs, hence 1 � tq is invertible inside Tq.De�ne then s � sqp1�tqq�1{2. By its de�nition, s is an isometry. By Coburn's


theorem, we �nd that we have a �-homomorphism

C�pSq Ñ Tq, S ÞÑ s.

As clearly πqpsq � S (under the isomorphism of Proposition 2.2.12), it followsthat πq is bijective.

Although the Tq are all C�-isomorphic, one would like to say Tq is morecommutative than Tq1 if q ¡ q1. To substantiate this, one would like to havea more natural way of comparing di�erent Tq when the parameter q varies.The notion of �eld of C�-algebras is ideally suited for this.

De�nition 2.2.14. Let X be a compact Hausdor� space. A CbpXq-algebrais a unital C�-algebra A together with a unital inclusion

π : CbpXq � ZpAq,

where ZpAq is the center of A.8

For x P X, the �ber C�-algebra Ax at x is de�ned as the quotient Ax � A{Jxwhere Jx is the closed 2-sided ideal in A generated by the functions in CbpXqwhich vanish at x P X. We call the canonical map πx : AÑ Ax the evaluationmap at x.

One can thus interpret a CbpXq-algebra A as consisting of anX-parametrizedcollection of C�-algebras Ax. A map x ÞÑ ax P Ax is then considered a`continuously varying' family of elements in the Ax if and only if there existsa P A with πxpaq � ax for all x P X. We say that the assignment x ÞÑ Axforms a �eld of C�-algebras.

Lemma 2.2.15. Let X be a compact space, and A a CbpXq-algebra. Thenfor all a P A, the map

x ÞÑ }πxpaq}is upper semi-continuous.

Proof. Let a P A. We are to prove that for any net xα Ñ x, one haslim supα }πxαpaq} ¤ }πxpaq}.

8We recall that the center of an algebra A is the set of z P A with zx � xz for allx P A.


Fix x P X. Take ε ¡ 0. By de�nition of the quotient norm, we can �nd�nitely many fi P CbpXq and ai P A such that fipxq � 0 for all i and suchthat, with t � °i fiai, one has

}πxpaq} ¥ }a� t} � ε.

Now for each δ ¡ 0, we can �nd an open neigborhood U of x and a functiong P CbpXq such that

a) }g} ¤ 1,

b) gpyq � 1 for y P U ,c) }gfi} δ for all i.

It then follows that }gt} ¤ δ°i }ai}. Hence by choosing δ small enough, we

can assume }gt} ¤ ε. But then

}πxpaq} ¥ }a� t} � ε

¥aq

}gpa� tq} � ε

¥}gt}¤ε

}ga} � 2ε

� }a� p1� gqa} � 2ε.

But since gpyq � 1 for all y P U , we have p1 � gqpyq � 0. So we conclude byde�nition of πy that

}πxpaq} ¥ }πypaq} � 2ε, @y P U.

This proves the upper semi-continuity.

In general however, there is no reason why the map x ÞÑ }πxpaq} should becontinuous.

Example 2.2.16. Consider X � r�1, 1s and Y � r1,8q. Let

A � CbpX � Y q.

Then A is a CbpXq-algebra by the map CbpXq Ñ CbpX � Y q associated tothe projection X�Y Ñ X. Let φ be a continuous bounded positive functionsuch that φpx, yq � 0 for x 0 and φpx, yq � 1 for x ¡ 0 and y ¡ 1

x(such


a function clearly exists). Then in the �ber Ax with x ¡ 0, we have thatπxpφq is a positive element with 1 inside its spectrum, hence }πxpφq} ¥ 1. Byupper semicontinuity, we conclude also }π0pφq} ¥ 1. But clearly }πxpφq} � 0for x 0. Hence x ÞÑ }πxpφq} is not lower semicontinuous in 0.

Hence, to avoid such pathological situations, we often impose that the abovenormfunction should be continuous.

De�nition 2.2.17. Let X be a compact Hausdor� space, and A a CbpXq-algebra. We say that the assignment x ÞÑ Ax forms a continuous �eld ofC�-algebras if and only if for each a P A, the map

x ÞÑ }πxpaq}

is continuous.

A general way to produce continuous CbpXq-algebras is the following.Proposition 2.2.18. Let X be a compact Hausdor� space. Let A be aCbpXq-algebra, and suppose there exist states ϕx on Ax such that

a) For all x, the GNS-representation πϕx of Ax is injective.

b) For all a P A, the function x ÞÑ ϕxpaxq is continuous.Then x ÞÑ Ax is a continuous �eld of C�-algebras.

Proof. As the πϕx are injective, it follows that they are isometric. As theGNS-vectors ξϕx are cyclic for Ax (and as the maps AÑ Ax are surjective),we have for all a P A and x P X that

}ax} � supt|xbxξϕx , πϕxpaxqcxξϕxy | b, c P A, }bxξϕx} ¤ 1, }cxξϕx} ¤ 1u� supt|ϕxpb�xaxcxq | b, c P A,ϕxpb�xbxq ¤ 1, ϕxpc�xcxq ¤ 1u.

Pick now x P X, a P A and 0 ε 1. Choose b, c P A with ϕxpb�xbxq ¤1, ϕxpc�xcxq ¤ 1 and

}ax} ¤ |ϕxpb�xaxcxq| � ε.

By continuity of x ÞÑ ϕxpfxq for all f P A, we can �nd an open neighborhoodU of x such that for all y P U , ϕypb�ybyq ¤ 1� ε, ϕypc�ycyq ¤ 1� ε and

}ax} ¤ |ϕypb�yaycyq| � 2ε.


Hence

}ax} ¤ }ay}p1� εq � 2ε

¤ }ay} � εp}a} � 2q.

It follows that x ÞÑ }ax} is lower semi-continuous, hence continuous byLemma 2.2.15.

Let us now show that the Toepliz algebras Tq can be collected into a �eld ofC�-algebras for 0 ¤ q ¤ 1, where we de�ne T1 � CbpDq (with D the closedunit disc). In T1, we de�ne s1 as the identity function z ÞÑ z, and we de�neT1 as the �-algebra generated by s1.

Proposition 2.2.19. Let I � r0, 1s. Let A be the unital �-algebra generatedby elements Q, T with Q� � Q, QT � TQ and

T �T �QTT � � 1�Q.

Then A admits a universal C�-envelope A with respect to the extra conditions}T } ¤ 1 and 0 ¤ Q ¤ 1.

Moreover, the assigment p ÞÑ ppQq on polynomials admits an extension to animbedding CbpIq Ñ A, and the associated CbpIq-algebra A satis�es Aq � Tq.

Proof. Since we impose norm restrictions on generators of A , it is clear thatthe universal C�-envelope A as above exists. Moreover, from the univeralrelations of A , we conclude that there exists a unital �-homomorphism

πq : AÑ Tq, Q ÞÑ q, T ÞÑ sq.

As 0 ¤ Q ¤ 1 in A, we also have a �-homomorphism

CbpIq Ñ A, f ÞÑ fpQq.

Since πqpfq � fpqq, we conclude that f ÞÑ fpQq is injective. Hence A is aCbpIq-algebra.Let now Iq � Kerpπqq. Then clearly fpQq P Iq if fpqq � 0. It follows thatJq � Iq, with Jq � A the ideal generated by functions in CbpXq vanishing atq. On the other hand, since Q� q P Jq, it follows that A{Jq is generated by


a single element Tq � T �Jq satisfying T �q Tq� qTqT �

q � 1� q. Hence we havea map

Tq Ñ A{Jq, sq ÞÑ Tq.

As this is clearly an inverse to the projection map πq : Aq Ñ Tq, we concludethat Iq � Jq.

We want to show that the �eld q ÞÑ Tq obtained from the CbpIq-algebra A iscontinuous. We will need some preparations.

De�nition 2.2.20. For 0 ¤ q 1, de�ne ϕq as the state

ϕq : Tq Ñ C, x ÞÑ p1� qq8̧

n�0

qnxen, xeny.

For q � 1, let ϕ1 be the state associated to the normalized Lebesgue measureon D.

Remark that ϕq is obviously well-de�ned since qn is absolutely summable. Itis then easily seen to be a state.

Lemma 2.2.21. Let 0 ¤ q ¤ 1 and x P Tq. Then

ϕqpsqxq � qϕqpxsqq.

Proof. For q � 1 this is obvious. For 0 ¤ q 1, we compute

ϕqpsqxq � p1� qq8̧

n�0

qnxen, sqxeny

� p1� qq8̧

n�0

qnxs�qen, xeny

� p1� qq8̧

n�1

p1� qnq1{2qnxen�1, xeny

� p1� qq8̧

n�0

p1� qn�1q1{2qn�1xen, xen�1y

� qp1� qq8̧

n�0

qnxen, xsqeny

� qϕqpxsqq.


Lemma 2.2.22. For 0 ¤ q 1 and m,n P N, we have

ϕqpsmq ps�q qnq � δm,nqn 1� q

1� qn�1.

We follow here the `convention' that 00 � 1.

Proof. First remark that from the de�ning relation s�qsq � qsqs�q � 1 � q, it

follows by induction that

ps�q qnsq � p1� qnqps�q qn�1 � qnsqps�q qn. (2.3)

Now from the de�nition of ϕq, it is clear that ϕqpsmq ps�q qnq � 0 when m � n.Write then an � ϕqpsnq ps�q qnq. From Lemma 2.2.21 and Equation (2.3), wededuce that

an � qϕqpsn�1q ps�q qnsqq

� qp1� qnqan�1 � qn�1an

Hence an � q 1�qn1�qn�1an�1. Since a0 � 1, we deduce an � qn 1�q

1�qn�1 .

Corollary 2.2.23. For a P A, the map q ÞÑ ϕqpaqq is continuous on r0, 1s.

Proof. By norm-density of A in A, and since }ϕq � πq} ¤ 1, it su�ces toprove the statement for a P A . Now from the commutation relations in A ,it follows easily that each element in A is a linear combination of productsQkT lpT �qm for k, l,m P N. It follows that it is su�cient to show that thefunctions

q ÞÑ ϕqpslqps�q qmqare continuous for l,m P N.

Now from Lemma 2.2.22, we see that ϕqpslqps�q qmq � δm,lqm 1�q

1�qm�1 for 0 ¤q 1. Clearly, this is a continuous function on r0, 1q. It remains to checkcontinuity in 1. But

ϕ1psl1ps�1qmq �1

π

»Dzlz̄mdz � δm,n

1

π

»D|z|2mdz � 1

m� 1.

Since limqÑ1 qm 1�q

1�qm�1 � 1m�1

, the corollary follows.


Theorem 2.2.24. The �eld of C�-algebras q ÞÑ Tq induced from the CbpIq-algebra A is continuous.

Proof. By combining Corollary 2.2.23 with Proposition 2.2.18, it is su�cientto see that the GNS-representations πϕq are injective on Tq for all 0 ¤ q ¤ 1.

Now for 0 q 1 and x P Tq, we have

xξϕq , πϕqpx�xqξϕqy � p1� qq¸nPN

qn}xen}2,

hence πϕqpxq � 0 implies x � 0.

A similar reasoning works for q � 1.

Finally, for q � 0, we have for all m,n ¥ 0, that

xsmq ξϕ0 , πϕ0pxqsnq ξϕ0y � ϕ0pps�0qmxsn0 q� xsm0 e0, xs

n0e0y

� xem, xeny.We conclude that also πϕ0 is injective.

2.2.4 More on representations of C�-algebras

Let A be a unital C�-algebra. We have seen that A admits an injective �-representation AÑ BpHq on a Hilbert space H. Let us consider the generalrepresentation theory of a C�-algebra in more detail.

Let us �rst present some general terminology.

De�nition 2.2.25. Let A be a unital C�-algebra, H1 and H2 two Hilbertspaces, and π1 and π2 unital �-representations on respectively H1 and H2.

An intertwiner between π1 and π2 is an operator x P BpH1,H2q such thatπ2paqx � xπ1paq for all a P A. We denote the set of all intertwiners by

Morpπ1, π2q � tx P BpH1,H2q | x intertwiner between π1 and π2u.

We say π1 is a subrepresentation of π2 if there exists an isometric intertwinerbetween π1 and π2.


We say π1 and π2 are unitarily equivalent if there exists a unitary intertwinerbetween π1 and π2.

Note that if x is an intertwiner from π1 to π2, then x� is an intertwiner from

π2 to π1. It is also clear that if x P Morpπ1, π2q and y P Morpπ2, π3q, thenyx P Morpπ1, π3q.Let us say that a closed subspace H1 � H is invariant under a representationπ of a C�-algebra A if πpAqH1 � H1.

De�nition 2.2.26. Let A be a unital C�-algebra. A unital �-representationπ of A on a Hilbert space H is called irreducible if H is non-zero and theonly invariant Hilbert spaces in H are t0u and H itself.

Remarks 2.2.27. 1. As π is a �-representation, one easily sees that H1 isinvariant if and only if its orthogonal complement is invariant. It followsthat a �-representation is irreducible if and only if it is atomic, i.e. cannot be written as a direct sum of two (non-trivial) �-representations.

2. If π is irreducible, any non-zero vector ξ P H is necessarily cyclic.

3. As we shall see later, every irreducible representation is even alge-braically irreducible: if π is irreducible and t0u � V � H is an invariantlinear subspace (not necessarily closed), then either V � t0u or V � H.

Lemma 2.2.28 (Schur's lemma for C�-algebra representations). Let A be aunital C�-algebra.

1. A unital �-representation π of A on a Hilbert space H is irreducible ifand only if Morpπ, πq � CidH.

2. If π1 and π2 are irreducible unital �-representations of A on respec-tive Hilbert spaces H1 and H2, then either Morpπ1, π2q � t0u or elseMorpπ1, π2q � Cu with u a unitary intertwiner.

Proof. Assume π is irreducible, and let x be an intertwiner from π to itself.We want to show that x is a scalar multiple of the identity operator.

Let us suppose �rst that we could �nd a normal intertwiner x which is nota scalar multiple of the identity operator. Then, as x� is also an intertwinerfrom π to itself, it follows by continuity that each element in C�pidH, xq is anintertwiner of π. Now as x is not a scalar, we can �nd, by functional calculus,two real nonzero functions f, g P CbpSppxqq such that fg � 0. As fpxq � 0,


the Hilbert space closure of fpxqH is a non-zero Hilbert space H1. Similarly,the Hilbert space closure of gpxqH is a non-zero Hilbert space H2. But since

xfpxqξ, gpxqηy � xξ, fpxqgpxqηy� xξ, pfgqpxqηy� 0,

H1 is orthogonal to H2.

It follows that t0u � H1 � H. But since fpxq P C�pidH, xq is an intertwiner,we have πpaqfpxqξ � fpxqπpaqξ for all a P A and ξ P H. We deduce bycontinuity that H1 is invariant. This is in contradiction with the assumptionthat π is irreducible.

Hence any normal intertwiner is scalar. Let us take now an arbitrary inter-twiner x. Then x�, x�x and xx� are also intertwiners. From the �rst partof the proof, it follows that x�x and xx� are scalars. As x�x and xx� arepositive operators, these scalars are positive. As x�x and xx� have the samenorm, we see that in fact these scalars are equal. It follows that x is normal(and in fact a positive multiple of a unitary). Then by the �rst part of theproof we conclude that x must necessarily be a scalar multiple of the identity.

This proves the �rst part of the proposition. To prove the second part,suppose x is a non-zero intertwiner between π1 and π2. By the �rst part ofthe lemma, it follows that x�x � xx� � λ for some λ ¡ 0. Hence u � λ�1{2xis a unitary intertwiner between π1 and π2.

If then y is another intertwiner from π1 to π2, the �rst part of the lemmagives that u�y � c for some c P C. Hence y � cu.

Assume now that ϕ is a state on a unital C�-algebra A. When is the asso-ciated GNS-representation πϕ irreducible? It turns out that this property isdetermined by the geometrical position of ϕ inside the state space.

De�nition 2.2.29. Let A be a unital C�-algebra, and SpAq its state space.We call ϕ P SpAq a pure state if ϕ is an extreme point of SpAq.Indeed, SpAq is clearly a compact convex set inside the linear space of contin-uous functionals on A (with the weak-� topology). Then, by de�nition, ϕ isan extreme point of SpAq if and only if it can not be written as tϕ1�p1�tqϕ2for two states ϕ1, ϕ2 with 0 t 1 and ϕ1 � ϕ2.


Theorem 2.2.30. Let ϕ be a state on a unital C�-algebra A. Then theGNS-representation πϕ is irreducible if and only if ϕ is extremal.

Proof. Assume that πϕ is not irreducible. Then H � H1 `H2 with H1 andH2 non-trivial orthogonal invariant subspaces. Let ξϕ � ξ1ϕ � ξ2ϕ be theorthogonal decomposition of ξϕ with respect to this direct sum. Then as ξϕis cyclic, it follows that neither ξ1ϕ or ξ2ϕ are zero. Writing t � }ξ1ϕ}2, we have}ξ2ϕ}2 � 1� t since ξϕ is a unit vector. In particular, 0 t 1. Put

ϕ1paq � 1

txξ1ϕ, πϕpaqξ1ϕy, ϕ2paq � 1

1� txξ2ϕ, πϕpaqξ2ϕy.

Then ϕ1 and ϕ2 are states, and ϕ � tϕ1 � p1 � tqϕ2. If we can show thatϕ1 � ϕ2, then we can conclude that ϕ is not pure. But if ϕ1 � ϕ2, thennecessarily ϕ1 � ϕ. This would mean that for all a P A,

xξϕ, πϕpaqξϕy � 1

txξ1ϕ, πϕpaqξ1ϕy

� 1

txξ1ϕ, πϕpaqξϕy.

By cyclicity of ξϕ, we deduce that ξϕ � 1tξ1ϕ, which gives a contradiction.

Conversely, suppose that πϕ is irreducible. Suppose that ϕ were not pure,say ϕ � tϕ1 � p1� tqϕ2 for 0 t 1 and states ϕ1, ϕ2 with ϕ1 � ϕ2. Then

ϕ1pa�aq � 1

ttϕ1pa�aq ¤ 1

tptϕ1pa�aq � p1� tqϕ2pa�aqq ¤ 1

tϕpa�aq.

It follows that we can de�ne a surjective bounded operator x P BpHϕ,Hϕ1qsuch that

x paξϕq � aξϕ1 , @a P A.As ξϕ is cyclic, it follows from this formula that x is an intertwiner. But asx�x is an intertwiner for πϕ, the irreducibility of πϕ gives that x�x is a scalarλ ¡ 0. Hence for all a P A,

ϕ1paq � xξϕ1 , aξϕ1y� xxξϕ, x paξϕqy� xξϕ, x�x paξϕqy� λxξϕ, aξϕy� λϕpaq.


As ϕ and ϕ1 are both states, this forces λ � 1 and hence ϕ � ϕ1. Bysymmetry, also ϕ � ϕ2, contradicting ϕ1 � ϕ2.

Let us show now that any C�-algebra has enough irreducible representations,giving a strengthening of the non-commutative Gelfand-Neumark theorem.

Theorem 2.2.31 (Gelfand-Raikov theorem9). Let A be a unital C�-algebraand a P A. Then there exists an irreducible unital �-representation π of Awith πpaq � 0.

Proof. Let χ be the character fpa�aq ÞÑ fp}a�a}q on B � C�p1A, a�aq. Asχ is a one-dimensional representation, it is clearly irreducible. Hence it is apure state on B. Now by Corollary 1.2.33, the restriction map SpAq Ñ SpBqis surjective. Hence the inverse image of χ forms a non-empty extremal facein the compact convex set SpAq. By the Krein-Milman theorem, there existsan extremal point ϕ of SpAq in the inverse image of χ. Then ϕ is a purestate which by construction satis�es ϕpa�aq � }a�a}. It follows that the GNS-representation of ϕ is an irreducible representation. Moreover, }πϕpaqξϕ}2 �ϕpa�aq � 0. Hence πϕpaq � 0.

For commutative C�-algebra, irreducible representations are easily classi�ed.

Proposition 2.2.32. Let A be a unital commutative C�-algebra. Then aunital �-representation π is irreducible if and only if π is one-dimensional.

In other words, all irreducible representations are characters.

Proof. Let π be an irreducible representation of A. Then for each a P A,the operator πpaq is itself an intertwiner of π. By Schur's lemma, πpaq isa scalar. As this holds for all a P A, it follows by irreducibility that π isone-dimensional.

On the other hand, each one-dimensional representation is necessarily irre-ducible.

9The actual Gelfand-Raikov theorem refers to the representation theory of locally com-pact Hausdor� groups, but as we will see later, it is a direct corollary of its C�-algebraversion.


We also have the following corollary concerning �elds of C�-algebras, sayingin a sense that elements in the global C�-algebra are determined by theirvalues in the �bers.

Corollary 2.2.33. Let X be a compact Hausdor� space, and A a CbpXq-algebra. Let a P A, and assume πxpaq � 0 for all x P X. Then a � 0.

Proof. Let a P A be non-zero. By the Gelfand-Raikov theorem, we can �ndan irreducible representation π of A with πpaq � 0. As π is irreducible,it follows that the self-intertwiner space of π is CidHπ . Hence π|ZpAq givesa character on ZpAq. In particular, there exists x P X such that πpfq �fpxqidHπ for all f P CbpXq. We conclude that π factors through Ax. Hence}πpaq} ¤ }πxpaq}, and πxpaq � 0.

From Proposition 2.2.32, it follows that for A a commutative unital C�-algebra, the set SppAq can be identi�ed with the set of all irreducible repre-sentations up to unitary equivalence. There is no obstruction to extend thisde�nition to the non-commutative case.

De�nition 2.2.34. Let A be a unital C�-algebra. We de�ne the set-theoreticspectrum of A as

SppAq � IrrpAq{ �,the set of all irreducible representations of A modulo unitary equivalence.

Unlike for commutative C�-algebras however, the spectrum SppAq can beill-behaved and retain very little of A itself. Nevertheless, SppAq can still beequipped with a compact (but not necessarily Hausdor�!) topology. Thistopology is the one inherited from a cruder kind of spectrum.

De�nition 2.2.35. Let A be a unital C�-algebra. A 2-sided ideal J of Ais called primitive if J � Kerpπq for some (non-zero) irreducible unital �-representation π. The primitive spectrum PrimpAq of A is the set of allprimitive ideals of A.

Remark 2.2.36. The primitive spectrum can be very small. Indeed, thereexist many simple C�-algebras, whose only 2-sided ideals are t0u and thewhole C�-algebra. For example, the Calkin algebra Q is simple.


Proposition 2.2.37. Let A be a unital C�-algebra. For I a closed 2-sidedideal of A, de�ne

VI � tJ P PrimpAq | I � Ju.Then

V � tVI | I a 2-sided closed idealuis the set of all closed subsets of a topology on PrimpAq.The above topology on PrimpAq is referred to as the Jacobson topology.

Proof. Cleary Vt0u � PrimpAq and VA � H. For Iα a collection of closed2-sided ideals, write I for the 2-sided ideal generated by all Iα. Then clearly

VI � XαVIα .

It remains to show that V is closed under �nite unions.

Let I, J be two closed 2-sided ideals. Then clearly VI Y VJ � VIXJ .

Conversely, take P P VIXJ . Let π be an irreducible representation withKerpπq � P . Then the closure of πpJqHπ is clearly an invariant subspace ofHπ, since J is a two-sided ideal. By irreducibility, one has

either πpJqHπ � 0 and hence πpJq � 0, or

πpJqHπ is dense in Hπ.

In the �rst case, we obtain J � P , hence P P VJ . In the second case,we deduce from I X J � P that πpxqπpyq � 0 for all x P I, y P J , henceπpIqπpJqHπ � t0u. By density of πpJqHπ in Hπ, this entails πpIq � 0, henceI � P and P P VI . In both cases, we obtain P P VI Y VJ .

Hence VI Y VJ � VIXJ .

De�nition 2.2.38. The topological spectrum of A is the set SppAq equippedwith the topology inherited from the Jacobson topology from PrimpAq underthe map

Ker : SppAq Ñ PrimpAq, rπs ÞÑ Kerpπq.

In other words, a subset of SppAq is open if and only if it is the inverse underKer of an open in PrimpAq.


Remark 2.2.39. In general, the map SppAq Ñ PrimpAq is not injective.Indeed, one can show that for any in�nite-dimensional separable simple C�-algebra the set SppAq is not a singleton.

Proposition 2.2.40. The topological spaces SppAq and PrimpAq are com-pact.

Proof. As Ker : SppAq Ñ PrimpAq is surjective, it is open by de�nition ofthe topology on SppAq. It hence su�ces to prove that PrimpAq is compact.

We use the following criterion for compactness: if Fα is any collection ofclosed subsets for which the intersection of any �nite subfamily is neverempty, then the intersection of all Fα is not empty.

For our situation, choose closed 2-sided ideals Iα and suppose that XαVIα �H. Then the closure of the C�-algebra generated by all Iα contains A, andin particular contains the unit of A. Now the C�-algebra generated by theIα is just the closure of the linear span of all Iα. Hence we can �nd �nitelymany αi and xi P Iαi such that }1 �°i xi} 1

2. In particular, we deduce

that°i xi is invertible. But this means that the 2-sided ideal generated by

the Iαi is already A, hence XiVIαi � H.

When A is a commutative unital C�-algebra, the space SppAq already carriedthe topology of weak�-convergence. Let us show that it coincides with thetopology inherited from PrimpAq. Note that for A commutative, one of coursehas that the map SppAq Ñ PrimpAq is bijective.Proposition 2.2.41. Let A be a commutative unital C�-algebra. Then a netof characters χα on A converges to a character χ for the weak�-topology ifand only if Kerpχαq converges to Kerpχq for the Jacobson topology.

Proof. Suppose �rst that χα Ñ χ for the weak�-topology. Then by de�nitionwe have Kerpχαq Ñ Kerpχq if and only if for any 2-sided closed ideal I ofA with I � Kerpχq, one has eventually I � Kerpχαq. But suppose that thiswere not the case. Then we can �nd a subnet χβ such that I � Kerpχβqfor all β. But then for each x P I, we have χpxq � limβ χβpxq � 0. HenceI � Kerpχq, a contradiction.

Hence the map Ker : SppAq Ñ PrimpAq is a continuous bijection. To show


that it is a homeomorphism, it thus su�ces to prove that PrimpAq is Haus-dor�.

Now by the Gelfand-Neumark theorem, we may identify A � CbpSppAqq(where SppAq has the weak�-topology). Take χ, χ1 P SppAq, and let U andU 1 be open neighborhoods of respectively χ and χ1 with U X U 1 � H. LetI be the closed 2-sided ideal of functions which vanish on U c, and similarlyI 1 the closed 2-sided ideal of functions which vanish on U 1c. Then VI �PrimpAq consists of all Kerpχ2q such that fpχ2q � 0 for all f P I. Hence,by the Urysohn lemma, VcI is an open neighborhood of Kerpχq, and similarlyVcI 1 is an open neighborhood of Kerpχ1q. Clearly this description also givesVcI X VcI 1 � H, as U X U 1 � H. Hence PrimpAq is Hausdor�.

As an example, let us examine the topological spectrum of the Toeplitz C�-algebra.

Lemma 2.2.42. Up to unitary equivalence, any irreducible �-representationof the Toeplitz C�-algebra T is either the standard representation of T onl2pNq, or else a character

χz : T Ñ C, S ÞÑ z

for some z P S1. These representations are all mutually inequivalent.

Proof. The fact that the above representations are mutually inequivalent isclear. It is also clear that all characters χz are irreducible.

Let us show that the standard representation T � Bpl2pNqq is irreducible.In fact if ξ, η in l2pNq, the rank one operator

Tξ,η : l2pNq Ñ l2pNq, ζ ÞÑ xξ, ζyηlies in B0pl2pNqq � T . It follows that any non-zero vector in l2pNq is cyclicfor T , hence the representation of T on l2pNq is irreducible.It remains to show that the representations in the statement of the lemmaexhaust all irreducible representations up to unitary equivalence.

But let π be an irreducible representation of T . From the Wold decomposi-tion appearing in Theorem 2.2.8, we may assume H � l2pX � Nq ` G withthen πpSq � SX ` u with u unitary. From the irreducibility of π, we con-clude that either G � t0u or X � H. In the �rst case, irreducibility implies


that X must be a singleton (otherwise we have a direct sum decompositionl2pX � Nq � l2pX1 � Nq ` l2pX2 � Nq as representations for any decompo-sition X � X1 \ X2), in which case this is just the ordinary representationon l2pNq. In the second case, π quotients to an irreducible representation ofCbpS1q, which is then necessarily a character. We deduce that in this caseπ � χz for some z P S1.

Corollary 2.2.43. The map

Ker : SppT q Ñ PrimpT q

is bijective.

Proof. If π is the standard representation, then Kerpπq � t0u. Hence Kerpπq �Kerpχzq for any character z. Since clearly Kerpχzq � Kerpχwq for z � w, thecorollary follows.

From the above, we can write SppT q � S1 Y t u, with representing thestandard representation on l2pNq. It remains to determine the topology ofSppT q.Proposition 2.2.44. A subset X � SppT q is closed if and only if X � SppT qor X � S1 is closed (for the ordinary topology on S1).

Proof. Under the identi�cation SppT q � PrimpT q, the point correspondsto the zero ideal t0u. But clearly t0u P VI for a closed 2-sided ideal I if andonly if I � t0u. It follows that SppAq is the only closed set containing .It now su�ces to show that χzα Ñ χz in SppT q if and only if zα Ñ z P S1.This follows along similar lines as in Proposition 2.2.41.

One may hence visualize the spectrum of T as being the closed unit disc withall points in the interior identi�ed into a `fat open point' whose closure is thewhole unit disc.

2.2 the abstract toeplitz algebra - vrije universiteit …homepages.vub.ac.be/~kdecomme/course/week...

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